Step by step
The question
Question: the probability of snow is 0.2.
If snow, the probability of late is 0.5.
If no snow, it is 0.1.
Find the probability of snow, given late.
1 Draw the tree
The branches from one point add up to 1.
So no snow is 1 minus 0.2: 0.8.
And not late is 1 minus each: 0.5 and 0.9.
2 Multiply along the branches
Snow then late: we multiply along the branches.
0.2 times 0.5 is 0.1.
The same for every end.
No snow then late: 0.8 times 0.1 is 0.08.
3 Add the paths
Late can happen two ways: after snow, or after no snow.
We add the two paths.
Careful!
Multiply along each branch first.
0.5 only counts when it is snow.
0.1 plus 0.08 is 0.18.
4 Divide by the given
Given late: only the two late paths count, 0.18.
Our path, snow then late, is 0.1.
0.1 divided by 0.18 is about 0.556, to 3 decimal places.
5 Check
Let's check: the four ends add up to 0.1 plus 0.1 plus 0.08 plus 0.72.
That is 1: every way is counted once.
So the answer is right!
Let's look back.
First: draw the tree.
Then: multiply along the branches.
Then: add the paths.
Then: divide by the given.
Last: check.